Let ${P}$ denote the set of all polynomial with variable $x\in [0,1]$, I need to know what is the closure of ${P}$ in $C[0,1]$?
Well, Stone-Weierstrass theorem says: If $f\in C[0,1]$ then there exists a sequence of polynomials $p_n(x)$ which converges uniformly to $f$. So can I say that $closure{P}=C[0,1]$?

2 Answers
2

Yes. The Stone-Weierstrass approximation theorem tells you that $P$ is dense in $C[0,1]$ with respect to $\|\cdot\|_\infty$ and a set $D$ is dense in a set $S$ if the closure of $D$ equals $S$. (by definition)

Talking about the definition of being dense. Can't we say that $[0,1)$ is dense in $[0,1)$?
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IlyaJul 19 '12 at 13:23

@Ilya Yes of course, the entire space is always dense in itself.
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Rudy the ReindeerJul 19 '12 at 13:26

@HenningMakholm Well in the given case the whole space is complete with respect to $\|\cdot\|_\infty$. But the definitions I've seen of "dense set" don't put any requirement on the space, in particular, they don't require the space to be complete.
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Rudy the ReindeerJul 19 '12 at 13:32